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ocr AS Level Further Mathematics A (Y535-01) Question Paper June2024.
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ocr AS Level Further Mathematics A (Y535-01) Question Paper June2024.
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ocrFriday 14 June 2024 – Afternoon
AS Level Further Mathematics A
Y535/01 Additional Pure Mathematics
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for AS Level
Further Mathematics A
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer
Booklet. If you need extra space use the lined pages at the end of the Printed
Answer Booklet. The question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be
given for using a correct method, even if your answer is wrong.
• Give non-exact numerical answers correct to 3 significant figures unless a different
degree of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value
is needed use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 4 pages.
ADVICE
• Read each question carefully before you start your answer.
Turn over
, 2
1 In this question you must show detailed reasoning.
The number N is written as 28A3B in base-12 form.
Express N in decimal (base-10) form. [2]
2 The points A and B have position vectors a and b relative to the origin O. It is given that
J N KJ ON
K2 b m , where m is a real parameter.
aO =
KK 4 OO and = K-4O
K O
3
L mP L 6P
(a) In the case when m = 3, determine the area of triangle OAB. [4]
(b) Determine the value of m for which a # b = 0. [2]
3 The surface S has equation
z = f(x, y), where f(x, y) = 4x2y - 6xy2 - 1 12x4 for all real values of x
and y. You are given that S has a stationary point at the origin, O, and a second stationary point at
the point P(a, b, c), where c = f(a, b).
(a) Determine the values of a, b and c. [6]
(b) Throughout this part, take the values of a and b to be those found in part (a).
(i) Evaluate fx at the points U1 ^a - 0.1, b , f(a - 0.1, b)h and U2 ^a + 0.1, b , f(a + 0.1, b)h.
[2]
(ii) Evaluate fy at the points V1 ^a , b - 0.1, f (a, b - 0.1)h and V2 ^a , b + 0.1, f(a, b + 0.1)h.
[2]
(iii) Use the answers to parts (b)(i) and (b)(ii) to sketch the portions of the sections of S,
given by
• z = f(x, b), for x - a G 0.1,
• z = f (a, y), y - G 0.1. [2]
for b
4 The first five terms of the Fibonacci sequence, "Fn,, where n H 1, are
F1 = 1, F2 = 1, F3 = 2, F4 = 3 and F5 = 5.
(a) Use the recurrence definition of the Fibonacci sequence, Fn +1 = Fn + Fn -1 , to express
Fn +4 in terms of Fn and Fn - 1 . [2]
(b) Hence prove by induction that Fn is a multiple of 3 when n is a multiple of 4. [3]
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